HOW (method 2 — peel off common factors one at a time):
Keep dividing top and bottom by any shared factor (2, 3, 5, …) until none remain. Slower but needs no GCD.
Cut a whole into b equal parts and take a of them.
Why must the denominator be non-zero?
You can't split a whole into 0 equal parts; division by 0 is undefined.
What are equivalent fractions?
Different fractions with the same value, e.g. 21=42=10050.
Rule to create an equivalent fraction?
Multiply (or divide) numerator AND denominator by the same non-zero number.
Why does multiplying top & bottom by k not change the value?
Because kk=1 and multiplying by 1 changes nothing.
When is a fraction in lowest terms?
When gcd(numerator, denominator) =1.
Fastest way to simplify?
Divide numerator and denominator by their GCD.
Cross-multiplication test for ba=dc?
True iff a×d=b×c.
Why can't you add the same number to top and bottom?
Adding changes the ratio; only ×/÷ by kk=1 preserves value.
Write 53 with denominator 20.
Multiply top & bottom by 4 → 2012.
Simplify 6045.
43 (÷5 then ÷3, or ÷15).
Recall Feynman: explain to a 12-year-old
Imagine a chocolate bar. If you break it into 2 big chunks and eat 1, you ate one-half. Now break the same bar into 4 pieces and eat 2 — you ate exactly the same chocolate! So "1 out of 2" and "2 out of 4" are the same amount, just counted with smaller pieces. Simplifying means going backwards: using the biggest possible pieces so the numbers are as small as they can be. And the golden rule: whatever you do to the top number, do to the bottom — but only times or divide, never plus, or you change how much chocolate you're really talking about.
Dekho, fraction ka matlab simple hai: kisi cheez ko barabar tukdon mein kaato aur kuch tukde lo. ba mein b batata hai kitne barabar tukde kiye, aur a batata hai kitne tukde tumne liye. Ab mazedaar baat: ek hi amount ko alag-alag numbers se likha ja sakta hai. Aadhi pizza ko tum 21 likho ya 42 — pizza to utna hi hai! Inhe equivalent fractions kehte hain.
Rule yaad rakho: numerator aur denominator dono ko same number se multiply ya divide karo, value nahi badlegi. Kyun? Kyunki kk=1 hota hai, aur kisi bhi cheez ko 1 se multiply karne se woh badalti nahi. Bas top-bottom dono ke saath same kaam karna zaroori hai — sirf top karoge to galat ho jaayega.
Simplify karna ulta process hai: fraction ko sabse chhote, saaf numbers mein laana. Uske liye numerator aur denominator ka HCF (GCD) nikaalo aur dono ko usse divide kar do. Jaise 2418 mein HCF 6 hai, to 24÷618÷6=43. Jab tak top-bottom mein koi common factor bacha ho, simplify adhoora hai.
Ek sabse badi galti se bacho: same number add mat karo top-bottom mein! 42 mein 1 jodkar 53 likhna galat hai (0.5 vs 0.6). Sirf times ya divide chalega. Aur equivalence check karne ka jugaad — cross-multiply: ba=dc tabhi jab a×d=b×c. Bas itna, fractions ke saare topics ki base yahi hai.
